Provable ICA with Unknown Gaussian Noise, and Implications for Gaussian Mixtures and Autoencoders

Abstract

We present a new algorithm for independent component analysis which has provable performance guarantees. In particular, suppose we are given samples of the form y=Ax+η where A is an unknown but non-singular n×n matrix, x is a random variable whose coordinates are independent and have a fourth order moment strictly less than that of a standard Gaussian random variable and η is an n-dimensional Gaussian random variable with unknown covariance Σ: We give an algorithm that provably recovers A and Σ up to an additive ϵϵ and whose running time and sample complexity are polynomial in n and 1/ϵ. To accomplish this, we introduce a novel “quasi-whitening” step that may be useful in other applications where there is additive Gaussian noise whose covariance is unknown. We also give a general framework for finding all local optima of a function (given an oracle for approximately finding just one) and this is a crucial step in our algorithm, one that has been overlooked in previous attempts, and allows us to control the accumulation of error when we find the columns of A one by one via local search.